sign in sign up
<<
Home , Euler class, Fiber (mathematics), Line bundle, Normal bundle, Section (fiber bundle), Vector bundle

LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES

1. Vector Bundles In general, smooth are very “non-linear”. However, there exist many smooth manifolds which admit very nice “partial linear structures”. For example, given any smooth M of n, the tangent

TM = {(p, Xp) | p ∈ M,Xp ∈ TpM} is “linear in tangent variables”. We have seen in PSet 2 Problem 9 that TM is a smooth manifold of dimension 2n so that the canonical projection π : TM → M is a smooth . A local chart of TM is given by −1 n T ϕ = (π, dϕ): π (U) → U × R , where {ϕ, U, V } is a local chart of M. Note that the local chart T ϕ “preserves” the −1 linear structure nicely: it maps the vector π (p) = TpM isomorphically to the {p} × Rn. As a result, if you choose another chart (ϕ, ˜ U,e Ve) containing p, −1 n n then the map T ϕe◦T ϕ : {p} ×R → {p} ×R is a linear isomorphism which depends smoothly on p. In general, we define Definition 1.1. Let E,M be smooth manifolds, and π : E → M a surjective smooth map. We say (π, E, M) is a of rank r if for every p ∈ M, there exits an open neighborhood Uα of p and a diffeomorphism (called the local trivialization) −1 r Φα : π (Uα) → Uα × R so that −1 r (1) Ep = π (p) is a r dimensional vector space, and Φα|Ep : Ep → {p} × R is a . (2) For Uα ∩ Uβ 6= ∅, there is a smooth map gβα : Uα ∩ Uβ → GL(r, R) so that −1 r Φβ ◦ Φα (m, v) = (m, gβα(m)(v)), ∀m ∈ Uα ∩ Uβ, v ∈ R . We will call E the total space, M the base and π−1(p) the fiber over p. (In the case there is no ambiguity about the base, we will denote a vector bundle by E for short.) (Roughly speaking, a vector bundle E over M is “a smooth varying family of vector spaces parameterized by a base manifold M”.) Remark. A vector bundle of rank 1 is usually called a .

Remark. It is easy to define conception of a vector sub-bundle: a vector bundle (π1,E1,M) is a sub-bundle of (π, E, M) if (E1)p ⊂ Ep for any p, and π1 = π|E1 .

1 2 LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES

Example. Here are some known examples: (1) For any smooth manifold M, E = M × Rr is a trivial bundle over M. (2) The TM and the T ∗M are both vector bundles over M. (3) Given any smooth submanifold X ⊂ M, the

NX = {(p, v) | p ∈ X, v ∈ NpX},

(where NpX is the quotient vector space TpM/TpX) is a vector bundle over X. Note: NX is NOT a vector sub-bundle of TM. (4) Any rank r distribution V on M is a rank r vector bundle over M. It is a vector sub-bundle of the tangent bundle TM.

n Example. The canonical line bundle over RP = {l is a line through 0 in Rn+1} is 1 n n+1 γn = {(l, x) | l ∈ RP , x ∈ l ⊂ R }. (Can you write down a local trivialization?) 1 1 1 In particular if n = 1, we have RP ' S . In this case the canonical line bundle γ1 is nothing else but the infinite M¨obiusband, which is a line bundle over S1. 1 1 Another way to obtain the infinite M¨obiusband γ1 is to identify S with [0, 1], with end points 0 and 1 “glued together”. Then the M¨obiusband is [0, 1] × R, with “boundary lines {0} × R and {1} × R glued together” via (0, t) ∼ (1, −t). Example. One can extend operations on vector spaces to operations on vector bundles. (1) Given any vector bundle (π, E, M), one can define the dual vector bundle by ∗ replacing each Ep with its dual Ep . (How to define local trivializations? What ∗ are the transition maps gβα’s?) For example, T M is the of TM. (2) Let (π1,E1,M) and (π2,E2,M) be two vector bundles over M of rank r1 and r2 respectively. Then the bundle (π1 ⊕ π2,E1 ⊕ E2,M) is the rank −1 r1 + r2 vector bundle over M with fiber (π1 ⊕ π2) (p) = (E1)p ⊕ (E2)p. (How to define local trivializations? What are the transition maps gβα’s?) (3) Then the product bundle (π1 ⊗ π2,E1 ⊗ E2,M) is the rank r1r2 vector −1 bundle over M with fiber (π1 ⊗ π2) (p) = (E1)p ⊗ (E2)p. (How to define local trivializations? What are the transition maps gβα’s?) For example, we have the (k, l)-tensor bundle ⊗k,lTM := (TM)⊗k ⊗ (T ∗M)⊗l over M. (4) Similarly one can define the exterior power bundle ΛkT ∗M, whose fiber at point k ∗ p ∈ M is the linear space Λ Tp M. (Local trivializations? Transition maps?) (5) Let f : N → M be a smooth map, and (π, E, M) a vector bundle over M. Then one can define a pull-back bundle f ∗E over N by setting the fiber over x ∈ N to be the fiber of Ef(x). (Local trivializations? Transition maps?) In particular, the restriction of a vector bundle (π, E, M) to a submanifold N of the base manifold M is a vector bundle over the submanifold N. (Note: this is not a vector sub-bundle of (π, E, M)!) LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES 3

We have seen that any smooth manifold can be embedded into the simplest mani- fold: an Euclidian space. A natural question is whether the same conclusion holds for vector bundles? More precisely, can we embed any vector bundle (π, E, M) into some trivial bundle M × RN as a vector sub-bundle? The answer is yes if M is compact, and the proof is similar to the proof of the “simple Whitney embedding theorem” in Lecture 9. (The conclusion could be wrong if M is non-compact.) Theorem 1.2. If M is compact, then any vector bundle E over M is isomorphic to a sub-bundle of a trivial vector bundle over M. Proof. (Please compare this proof with the proof of Theorem 1.1 in Lecture 9). Take a finite open cover {Ui}1≤i≤k of M so that E is trivial over each Ui via the trivialization 2 −1 r maps Φi = (π, Φi ): π (Ui) → Ui × R . Let {ρi}1≤i≤k be a P.O.U. subordinate to {Ui}1≤i≤k. Consider r k 2 2 Φ: E → M × (R ) , v 7→ (π(v), ρ1(π(v))Φ1(v), ··· , ρk(π(v))Φk(v)).

Note that on each fiber Ep, Φ is a linear isomorphism onto its . The conclusion follows, since the image of Φ is a vector sub-bundle of M × Rrk: for any p ∈ M, there 2 exists i so that ρi(p) 6= 0. Then the map Φi induces a local trivialization near p.  As a corollary we get the following fundamental fact in topological K-theory: Corollary 1.3. If M is compact, then for any vector bundle E over M, there exists a vector bundle F over M so that E ⊕ F is a trivial bundle over M.

Proof. We have seen that E is a vector sub-bundle of a trivial bundle M × RN over N M. Now we put an inner product on R , and take the fiber Fp of F at p ∈ M to be N the orthogonal complement of Ep in R . (One should check that F is a vector bundle over M.) 

2. Sections of vector bundles The following definition is natural: Definition 2.1. A (smooth) of a vector bundle (π, E, M) is a (smooth) map s : M → E so that π ◦ s = IdM . The of all sections of E is denoted by Γ(E), and the set of all smooth sections of E is denoted by Γ∞(E). ∞ Remark. Obviously if s1, s2 are smooth sections of E, so is as1 + bs2. So Γ (E) is an (infinitely dimensional) vector space. In fact, one can say more: if s is a smooth section of E and f is a smooth on M, then fs is a smooth section of E. So Γ∞(E) is a C∞(M)-. (According to the famous Serre-Swan theorem, there is an equivalence of categories between that of finite rank vector bundles over M and finitely generated projective modules over C∞(M).) Remark. Many geometrically interesting objects on M are defined as smooth sections of some (vector) bundles over M. For example, 4 LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES

• A smooth vector field on M = a smooth section of TM. • A smooth (k, l)-tensor field on M = a smooth section of ⊗k,lTM. • A smooth k-form on M = a smooth section of ΛkT ∗M. • A on M = a non-vanishing smooth section of ΛnT ∗M. • A Riemannian metric on M = a smooth section of ⊗2T ∗M satisfying some extra (symmetric, positive definite) conditions • A symplectic form on M = a smooth section of Λ2T ∗M satisfying some extra (closed, non-degenerate) conditions By definition any vector bundle admits a trivial smooth section: the zero section

s0 : M → E, p 7→ (p, 0). On the other hand, it is possible that a vector bundle admits no non-vanishing section. For example, we have seen • M is orientable if and only if ΛnT ∗M admits a global non-vanishing section. • TS2n admits no non-vanishing section. Here is another example: 1 n Example. Consider the canonical line bundle γn over RP . We claim that there is no n 1 non-vanishing smooth section s : RP → γn. To see this we consider the composition n P n s 1 ϕ : S −→ RP −→ γn, where P is the projection p(±x) = lx, the line through x. By definition, ϕ is of form

x 7→ (lx, f(x)x), where f is a smooth function on Sn satisfying f(−x) = −f(x). By mean value property, f vanishes at some x0. It follows that s vanishes at x0 also. Although there might be no non-vanishing global sections, locally there are plenty of non-vanishing sections. Let E be a rank r vector bundle over M, and U an open set in M.

Definition 2.2. A local frame of E over U is an ordered r-tuple s1, ··· , sr of smooth section of E over U so that for each p ∈ U, s1(p), ··· , sr(p) form a basis of Ep. Example. Let M be a smooth manifold and U be a coordinate patch. Then

• ∂1, ··· , ∂n form a local frame of TM over U. • dx1, ··· , dxn form a local frame of T ∗M over U. The following fact is basic. We will leave the proof as an exercise. Proposition 2.3. Let E be a smooth vector bundle over M. (1) A section s ∈ Γ(E) is smooth if and only if for any p ∈ M, there is a neighbor- hood U of p and a local frame s1, ··· , sr of E over U so that s = c1s1 +···+crsr for some smooth functions c1, ··· , cr defined in U. (2) E is a trivial bundle if and only if there exists a global frame of E on M. LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES 5

3. De Rham groups of vector bundles We have seen k r k k r k−r HdR(M × R ) ' HdR(M) and Hc (M × R ) ' Hc (M). A vector bundle E can be viewed as a “twisted product” of a smooth manifold M (the base) with a vector space (the fiber). So it is natural to study the relation between the cohomology groups of E and the cohomology groups of M. Proposition 3.1. For any vector bundle E over M, one has k k HdR(E) = HdR(M), ∀k. Proof. This is a consequence of the invariance: E is homotopy equivalent to M, since if we let s0 : M → E be the zero section, then π ◦ s0 = IdM , and s0 ◦ π ∼ IdE via the homotopy F : E × R → E, (x, v, t) 7→ (x, tv).  In general, the same result fails for compact supported cohomology groups. For example, let E be the infinite M¨obiusband, which is a line bundle over S1. Since E is non-orientable, one has 2 1 1 1 1 Hc (E) = 0 6' R ' HdR(S ) = Hc (S ). On the other hand, if we assume orientablity, then Proposition 3.2. Let E be a rank r vector bundle over M. Assume both E and M are oriented with finite dimensional compact supported de Rham cohomology groups, then k k−r Hc (E) ' Hc (M), ∀k. Proof. We apply Poincar´eduality twice: k n+r−k n+r−k k−r Hc (E) ' HdR (E) ' HdR (M) ' Hc (M).  Now let M be an oriented, compact and connected smooth n-manifold, and E an oriented vector bundle of rank r over M. The constant function 1 on M gives us 0 a degree-0 cohomology class [1] ∈ HdR(M). So the isomorphism above produces an r [T ] ∈ Hc (E). Definition 3.3. We call [T ] the Thom class of (π, E, M).

k k−r Remark. The isomorphism Hc (E) ' Hc (M) is called the Thom duality. It is given explicitly by an integration along the fibers map k k−r π∗ :Ωc (E) → Ω (M). 6 LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES

More precisely, on a local trivialization we can pick coordinates x1, ··· , xn of M and s1, ··· , sr of the fiber. For any k-form (k ≥ r) ω on M, locally we can write ω = f(x, s)(π∗θ) ∧ ds1 · · · ∧ dsr + terms with less fiber 1-forms, where θ is a (k − r)-form on M, and f is compactly supported. Then Z 1 r π∗ω := θ f(x, s) ds ··· ds . Rr One can check that

• π∗ω is well-defined. – To prove the independence of the choices of the fiber variables s1, ··· , sr, one need the following fact: Since E and M are both orientable, one can + always choose fiber variables so that the transition maps gαβ ∈ GL (n, R). k k−c • dπ∗ = π∗d. (=⇒ π∗ induces a linear map π∗ : Hc (E) → Hc (M).) k k−c • The induced map π∗ : Hc (E) → Hc (M) is an isomorphism. • Moreover, one has the projection formula ∗ ∗ ∗ π∗(π θ ∧ ω) = θ ∧ π∗ω, ∀θ ∈ Ω (M) and ω ∈ Ωc (E). The Thom class gives us the “inverse” to the “integration along fibers map”. More precisely, by definition, π∗([T ]) = 1. By using the projection formula above, one has −1 ∗ ∗ (π∗) ([ω]) = [π ω] ∪ [T ], ∀ω ∈ Zc (E). Let E be an oriented vector bundle of rank r over an oriented connected compact smooth manifold M. Let s : M → E be any global section of E. Then by using the pull-back, we get an element ∗ r s ([T ]) ∈ HdR(M). Proposition 3.4. The de Rham cohomology class s∗([T ]) is independent of the choices of s.

Proof. Let s0 : M → E be the zero section. Then s0 ∼ s via the homotopy F : M × R, (m, t) 7→ (m, ts(m)). ∗ ∗ So s ([T ]) = s0([T ]).  Definition 3.5. The de Rham cohomology class ∗ r χ(E) := s ([T ]) ∈ HdR(M) is called the of E.

The Euler class of a vector bundle is an obstruction to the existence of a non- vanishing global section. Theorem 3.6. If an oriented vector bundle E over an oriented compact connected smooth manifold M admits a non-vanishing global section, then χ(E) = 0. LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES 7

r Proof. Let s : M → E be an non-vanishing section. Take T ∈ Zc (E) so that [T ] is the Thom class. Take c ∈ R so that the section s1 = cs does not intersect supp(T ). It follows ∗ ∗ χ(E) = s1([T ]) = [s1T ] = 0.  Finally we remark that for E = TM the tangent bundle, one has χ(TM) = χ(M)[µ], n R where µ ∈ Ω (M) is any top form with M µ = 1, and n X i i χ(M) = (−1) dim HdR(M) i=0 is the number of M. This explains the name “Euler class”. More- over, this explains why any smooth vector field on S2n admits at least one zero, while there exists non-vanishing smooth vector fields on S2n+1: χ(S2n) 6= 0 while χ(S2n+1) = 0.